Completion in Operads via Essential Syzygies

Abstract

We introduce an improved Grobner basis completion algorithm for operads. To this end, we define operadic rewriting systems as a machinery to rewrite in operads, whose rewriting rules do not necessarily depend on an ambient monomial order. A Grobner basis of an operadic ideal can be seen as a confluent and terminating operadic rewriting system; thus, the completion of a Grobner basis is equivalent to the completion of a rewriting system. We improve the completion algorithm by filtering out redundant S-polynomials and testing only essential ones. Finally, we show how the notion of essential S-polynomials can be used to compute Grobner bases for syzygy bimodules. This work is motivated by the computation of minimal models of associative algebras and symmetric operads. In this direction, we show how our completion algorithm extends to the case of shuffle operads.